Vol. 84, No. 1, 1979

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The hyperbolic metric and coverings of Riemann surfaces

C. David (Carl) Minda

Vol. 84 (1979), No. 1, 171–182

Suppose X and Y are Riemann surfaces which have the open unit ball as universal covering surface. Let X,dσY be the hyperbolic metric on X, Y , respectively. Given any analytic function f : X Y the principle of hyperbolic metric asserts that (f(Y )∕dσX)(p) 1 for each point p X where f(Y ) is the pull-back to X via f of the hyperbolic metric on Y . Moreover, equality holds if and only if f is an (unbranched, unlimited) covering of X onto Y . This paper has two main objectives. The first is to study how the principle of hyperbolic metric can be strengthened if we only consider analytic functions which are not coverings. The second is to investigate the set of all analytic coverings of X onto Y .

Mathematical Subject Classification 2000
Primary: 30F20
Secondary: 57M12
Received: 5 December 1978
Published: 1 September 1979
C. David (Carl) Minda